All Questions
Tagged with cv.complex-variables reference-request
271 questions
5
votes
1
answer
510
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The space $H(D)$ of holomorphic functions.
A very natural example of a nuclear Montel space is the space $H(D)$ of all holomorphic functions on the open disc topologized by the family of seminorms
$$p_n(f)=\sup\{|f(z)|\colon |z|\leq 1-\tfrac{...
- 51
2
votes
1
answer
583
views
Brieskorn's proof of a theorem by Milnor about the Milnor number
I am looking for a reference or short explanation of a proof by E. Brieskorn.
In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in ...
- 1,124
11
votes
1
answer
676
views
Analysis and finitely generated groups
Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull.
So let $G$ be a finitely generated group and choose some finite set of generators. This allows to ...
- 8,098
3
votes
0
answers
178
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One-parameter groups acting on dual Banach spaces
Let $E$ be a Banach space, and $M=E^*$ (my application has $M$ a von Neumann algebra, but this is unimportant). Let $(\sigma_t)$ be a SOT cts one-parameter group on $E$: so for $t\in\mathbb R$, we ...
- 18.7k
7
votes
3
answers
1k
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Compactness properties of plurisubharmonic functions
I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.
Let $\...
- 60.5k
8
votes
3
answers
2k
views
Harmonic level sets and boundary data
This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great:
Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\...
1
vote
1
answer
554
views
Conformal mapping of C \ D* onto C \ (-1, 1) [closed]
Which is the concrete formula for the conformal mapping (normalized at infinity),
acting from $\mathbb C \backslash D^*$ onto
$\mathbb C\backslash[-1, 1]$?
Here $\mathbb C$ denotes the set of all ...
- 21
3
votes
1
answer
499
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methods for interpolating a function, holomorphic in the upper halfplane
Let $n,k\colon\mathbb{R}\to\mathbb{R}$ be real functions such that function $N$ given by $N(x)=n(x)-ik(x)$ is a holomorphic function in the upper half-plane. Also I know some additional properties of ...
- 1,284
3
votes
1
answer
2k
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Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)
The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....
- 1,795
2
votes
3
answers
632
views
How to find the almost period of an exponential polynomial
Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...
- 1,795
2
votes
1
answer
2k
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The normal derivative of the Green's function
I was wondering if anything was known about the following:
Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ ...
- 2,299
13
votes
4
answers
1k
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"Simple" Kahler manifolds
I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
- 4,343
15
votes
1
answer
2k
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Dirichlet series expansion of an analytic function
Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}...
- 7,127
1
vote
2
answers
698
views
Extension of harmonic function at infinity
Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property ...
- 1,795
4
votes
0
answers
715
views
some questions about properties of harmonic measure
The original post
The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ...
- 1,795
6
votes
1
answer
2k
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Approximation by analytic functions
Dear all.
Let
$$
f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I ...
- 3,343
21
votes
5
answers
7k
views
References for complex analytic geometry?
I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc....
212
votes
52
answers
82k
views
Ways to prove the fundamental theorem of algebra
This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...
31
votes
11
answers
13k
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Uniformization theorem for Riemann surfaces
How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
8
votes
1
answer
638
views
Composite residues with determinant denominators
I am looking for a good reference on composite residues of multi-variable contour integrals (something better and more explicit than Griffiths and Harris or Tsikh). This means I want to evaluate $\...
- 81
6
votes
4
answers
2k
views
Space of $(1,0)$-holomorphic forms on a Riemann surface
In a complex analysis course I have been given the following definition:
Let $X$ be a Riemann surface, denote by $H(1,0)$ the space of all $(1,0)$-holomorphic forms on $X$ and consider the quotient ...
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